Ecology 8310 Population (and Community) Ecology Seguing into from populations to communities Species interactions Lotka-Volterra equations Competition Adding in resources

Species interactions: Competition(-, -) Predation(+, -) (Herbivory, Parasitism, Disease) Mutualism(+, +) None(0, 0)

Species interactions: 1 a 11 a 14 =0 4 a 12 2 a ij >0 a ij <0 a a 16 5 a 15 a ij gives the per capita effect of species j on species i’s per capita growth rate, dN i /N i dt

Generalized Lotka- Volterra system: Special cases: 1.Exponential model: all a’s=0 2.Logistic model: a ii <0; others =0

N2N2 N1N1 dN 1 /N 1 dt Slope = a 12 Slope = a 11

N2N2 N1N1 dN 1 /N 1 dt Slope = a 12 Slope = a 11

Species interactions: 1 a 11 a 12 2 a ij >0 a ij <0 a 13 3 a 31 4 a 21 a 44 What can you say about the interactions between these species? Which are interspecific competitors? Which are predator and prey? Which are mutualists? Which show self limitation? a 43 a 34

Competition: 1 a 12 2 a 21 a 11 a 22 Alternate terminology:  ij = a ij /a ii, the effect of interspecific competition relative to the intraspecific effect (e.g., how many of species i does it take to have the same effect as 1 individual of species j?) Arises when two organisms use the same limited resource, and deplete its availability (intra. vs. interspecific) 12 R

Competition: 1 a 12 2 a 21 a 11 a 22

1 a 12 2 a 21 a 11 a 22 Can we use this model to understand patterns of competition among two species (e.g., coexistence and competitive exclusion)? E.g., Paramecium experiments by Gause… Competition:

Classic studies of resource competition by Gause (1934, 1935) Paramecium aurelia Paramecium bursaria Paramecium caudatum

Competitive exclusion: P. aurelia excludes P. caudatum

Paramecium caudatum Paramecium bursaria In contrast… Why this disparity, and can we gain insights via our model?

1 a 12 2 a 21 a 11 a 22 Competition:

At equilibrium, dN/Ndt=0: Competition:

N1N1 N2N2 Phase planes: K 1 /  12 Graph showing regions where dN/Ndt=0 (and +, -); used to infer dynamics Species 1’s zero growth isocline… dN 1 /N 1 dt=0 K1K1

N1N1 N2N2 Phase planes: K 1 /  12 What if the system is not on the isocline. Will what N 1 do? dN 1 /N 1 dt=0 K1K1

N1N1 N2N2 Phase planes: K2K2 dN 2 /N 2 dt=0 K 2 /  21

N1N1 N2N2 Phase planes: K 1 /  12 Putting it together… dN 1 /N 1 dt=0 K1K1 dN 2 /N 2 dt=0 Species 2 “wins”: N 2 * =K 2, N 1 * =0 (reverse to get Species 1 winning) K 2 /  21 K2K2

N1N1 N2N2 Phase planes: K 1 /  12 Your turn…. For A and B: 1)Draw the trajectory on the phase-plane 2)Draw the dynamics (N vs. t) for each system. dN 1 /N 1 dt=0 K1K1 dN 2 /N 2 dt=0 K 2 /  21 K2K2 A B

N1N1 N2N2 Phase planes: K 1 /  12 Another possibility… dN 1 /N 1 dt=0 K1K1 dN 2 /N 2 dt=0 “It depends”: either species can win, depending on starting conditions K 2 /  21 K2K2

N1N1 N2N2 Phase planes: K 1 /  12 dN 1 /N 1 dt=0 K1K1 K 2 /  21 K2K2 Your turn…. Draw the dynamics (N vs. t) for the system that starts at: Point A Point B A B

N1N1 N2N2 Phase planes: K 1 /  12 dN 1 /N 1 dt=0 K1K1 K 2 /  21 K2K2 Now do it for many starting points: Separatrix or manifold

N1N1 N2N2 Phase planes: K 1 /  12 A final possibility… dN 1 /N 1 dt=0 K1K1 dN 2 /N 2 dt=0 Coexistence! K 2 /  21 K2K2

N1N1 N2N2 Phase planes: K 1 /  12 “Invasibility”… dN 1 /N 1 dt=0 K1K1 dN 2 /N 2 dt=0 Mutual invasibility  coexistence! Why: because each species is self-limited below the level at which it prevents growth of the other K 2 /  21 K2K2

N1N1 N2N2 Invasibility: K 1 /  12 Contrast that with… dN 1 /N 1 dt=0 K1K1 dN 2 /N 2 dt=0 Neither species can invade the other’s equilibrium (hence no coexistence). K 2 /  21 K2K2

N1N1 N2N2 Coexistence: K 1 /  12 dN 1 /N 1 dt=0 K1K1 dN 2 /N 2 dt=0 K 2 /  21 K2K2

Coexistence: “intra > inter” Coexistence requires that the strength of intraspecific competition be stronger than the strength of interspecific competition.  Resource partitioning  Two species cannot coexist on a single limiting resource

Can we now explain Gause’s results? Paramecium aurelia Paramecium bursaria Paramecium caudatum Bacteria in water column Yeast on bottom

Resources: But what about resources? (they are “abstracted” in LV model) Research by David Tilman

Resources: Followed population growth and resource (silicate) when alone: Data = points. Lines = predicted from model

Resources: What will happen when growth together: why?

Resources: R*: resource concentration after consumer population equilibrates (i.e., R at which Consumer shows no net growth) Species with lowest R* wins (under idealized scenario: e.g., one limiting resource). If two limiting resources, then coexistence if each species limited by one of the resources (intra>inter): trade-off in R*s.

Next time: Tilman's R* framework